Biophysical Journal Volume 104 February 2013 533–540 533

Simple Moment-Closure Model for the Self-Assembly of Breakable AmyloidFilaments

Liu Hong* and Wen-An YongZhou-Pei Yuan Center for Applied Mathematics, Tsinghua University, Beijing, P. R. China

ABSTRACT In this work, we derive a simple mathematical model from mass-action equations for amyloid fiber formation thattakes into account the primary nucleation, elongation, and length-dependent fragmentation. The derivation is based on the prin-ciple of minimum free energy under certain constraints and is mathematically related to the partial equilibrium approximation.Direct numerical comparisons confirm the usefulness of our simple model. We further explore its basic kinetic and equilibriumproperties, and show that the current model is a straightforward generalization of that with constant fragmentation rates.

INTRODUCTION

As a typical self-assembly phenomenon, the amyloid fibrousaggregation caused by protein misfolding or (partially) un-folding has been proved to be an intrinsic feature of manydifferent kinds of proteins. It can occur in various bio-systems (cells and tissues) and is directly related to severaltypes of well-known neuron-degenerative diseases (1–3),including Alzheimer’s, Parkinson’s, and prion diseases.Thus, elucidating the fibrillation mechanisms of amyloidproteins will not only help us gain a better understandingof how and why amyloidosis arises and proceeds in vivo(4,5), it will also aid in the medical diagnosis and treatmentof amyloid-related diseases, and drug development (6).

Among the various experimental, computational, andtheoretical methods used in this field, modeling based onmass-action equations appears to be a successful approach(7,8). In this approach, the basic processes of fiber formationare modeled by chemical reactions with empirical reactionrates, and the time evolution of each chemical component(amyloid proteins or fibrils) is formulated according to thelaw of mass action. Mass-action equations have been widelyused due to their special advantages for quantitatively char-acterizing the kinetic procedure of amyloid fiber formation,theoretically relating various kinetic quantities (e.g., lagtime and apparent fiber growth rate) with model parameters(e.g., protein concentration and reaction rate constant), anddirectly interpreting experimental data with high precision.

The first notable adaptation of the above approach to thisfield was made in 1959 by Oosawa et al. (9), who investi-gated how native-state G-actin proteins transform intoF-actin. In Oosawa’s model, the conformational transitionbetween ordinary polymers and helical polymers was recog-nized as a key step for actin growth. The importance ofprotein conformational transition in fiber formation wassubsequently confirmed for many other amyloid proteins(10,11). In 1971, Eisenberg (12) proposed the mechanism

Submitted August 7, 2012, and accepted for publication December 21, 2012.

*Correspondence: zcamhl@tsinghua.edu.cn

Editor: Gijsje Koenderink.

� 2013 by the Biophysical Society

0006-3495/13/02/0533/8 $2.00

of subsequent monomer addition for the polymerization ofglutamate dehydrogenase. In 1974, Hofrichter et al. (13)combined the classical homogeneous nucleation (or primarynucleation) with subsequent monomer addition in a studyof sickle-cell hemoglobin gelation. This provided the basicmodeling framework for most subsequent studies (14,15). Inaddition, Ferrone et al. (16) suggested a new heterogeneousnucleation mechanism (monomer-dependent secondarynucleation) to account for the extreme autocatalysisphenomenon observed in the sickle-cell hemoglobin gela-tion induced by photolysis. In contrast to homogeneousnucleation, heterogeneous nucleation occurs on the surfaceof existing fibrils and thus depends on the concentrationsof both monomeric proteins and fibrils (17,18). In 1975,Oosawa and Asakura (7) introduced further steps of filamentfragmentation and association for breakable amyloid fila-ments, which can effectively produce new fiber seeds inthe absence of monomeric proteins. Recently, quantitativeanalysis of this monomer-independent secondary nucleationhas received much attention (8,19,20) and is the main focusof this work. In addition, many researchers have made greatcontributions to our understanding of random polymeriza-tion (21), on- and off-pathway competition (22), autocata-lytic surface growth (23), branching (24), and lateralassociation (25,26). These works have largely enhancedour understanding of amyloid fiber formation and revealedimportant connections among theoretical modeling, experi-mental data fitting, and prediction.

Although they have shown great success in modelingand application, mass-action equations suffer from anintrinsic bottleneck. In principle, the models involve infi-nitely many equations if we distinguish filament specieswith different lengths. Even for the realistic cases, thenumber of the species is quite large (from thousands tohundreds of thousands). Faced with such a high-dimensionalsystem of ordinary differential equations (ODEs), it is defi-nitely difficult to perform direct calculations or analyses.

An often-adopted method is to define some macro-measurable statistical quantities, such as number

http://dx.doi.org/10.1016/j.bpj.2012.12.039

534 Hong and Yong

concentration PðtÞ (zeroth-order moment of filament length)and mass concentration MðtÞ (first-order moment of fila-ment length) of filaments, and then derive their time-evolu-tion equations from the original mass-action equations (8).(Similar ideas have been widely applied to other problems,such as turbulence and neuron networks (27).) If the time-evolution equations are closed (e.g., for models withprimary nucleation and elongation, the time-evolution equa-tions for PðtÞ and MðtÞ can be shown as closed if boundaryterms are neglected), we can obtain the desired answer.However, if high-order moments or more complicated fibril-lation mechanisms (such as fragmentation) are considered,it seems impossible for the resulting equations to be closed.

To deal with this problem, in this work we introducea general moment-closure method (28). Precisely, we definea free energy for systems in a model that includes primarynucleation, elongation, and fragmentation. By taking theminimization of the free energy under proper constraints,we derive a closed system of two time-evolution equationsfor number concentration PðtÞ and mass concentrationMðtÞ of filaments. Numerical comparisons and direct fittingof experimental data show that this simple system is a quitegood approximation of the original mass-action equations.We also point out several basic kinetic properties of thetwo-equation system that may be useful for further studiesand applications.

MATERIALS AND METHODS

Modeling the kinetic processes of amyloid fiberformation

Mass-action equations

To quantitatively account for the formation of breakable amyloid filaments,

the general model under consideration (8) includes three basic processes–

primary nucleation, elongation, and fragmentation (see Fig. 1):

ncA1#kþn

k�nAnc ;

A1 þ Ai#kþe

k�eAiþ1; ðiRncÞ

Aiþj#kþfði;jÞ

k�fði;jÞ

Ai þ Aj; ði; jRncÞ

(1)

Biophysical Journal 104(3) 533–540

where the critical nucleus size is ncR2;A1 stands for monomeric proteins;

Ai are filaments of size i; and kþn ; kþe ; k

þf ði; jÞ and k�n ; k

�e ; k

�f ði; jÞ are the

forward and backward reaction rate constants for fiber nucleation, elonga-

tion, and fragmentation, respectively.

According to the law of mass action, the time-evolution equations for

½Ai�,the molar concentration of filament of size i, can be written as

d

dt½A1�¼ � nck

þn ½A1�ncþnck

�n ½Anc � � 2kþe ½A1�

XNj¼ nc

�Aj

�

þ 2k�eXN

j¼ ncþ1

�Aj

�;

d

dt½Ai�¼ 2kþe ½A1�ð½Ai�1� � ½Ai�Þ � 2k�e ð½Ai� � ½Aiþ1�Þ

þ 2XN

j¼ ncþi

kþf ði; j � iÞ�Aj

��Xi�nc

j¼ nc

kþf ðj; i� jÞ½Ai�

� 2XNj¼ nc

k�f ði; jÞ½Ai��Aj

�þXi�nc

j¼ nc

k�f ðj; i� jÞ�Aj

��Ai�j

�

þ �kþn ½A1�nc�k�n ½Ai� � 2kþe ½A1�½Ai�1�

þ 2k�e ½Ai��di;nc : ðiRncÞ

8>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

(2)

This is our mass-action equation. Thanks to the conservation law of mass,

we also have ½A1� þPN

i¼nci$½Ai� ¼ mtot: Note that similar equations can be

found elsewhere (8,14), but backward reactions were usually neglected for

simplicity in previous works.

Moment-closure method

It is easy to see that for the above choices of kþf ði; jÞ and k�f ði; jÞ,Eq. 2 generally cannot lead to closed equations for PðtÞ ¼PN

i¼nc½Ai� and

MðtÞ ¼PNi¼nc

i$½Ai�.To solve this problem, we adopt the moment-closure method (28) in the

kinetic theory. For this purpose, we construct the free energy function as

F ¼ εn

XNi¼ nc

½Ai� þ εe

XNi¼ nc

ði� ncÞ$½Ai�

þ kBT

"ð½A1� ln½A1� � ½A1�Þ þ

XNi¼ nc

ð½Ai� ln½Ai� � ½Ai�Þ#:

(3)

FIGURE 1 Illustration of primary nucleation,

elongation, and fragmentation processes in the

formation of breakable amyloid filaments.

Moment-Closure Model for Amyloid Fiber 535

where εn>0 represents the free energy associated with the nuclei region of

a fiber (i.e., the boundary energy penalty), εe<0 captures the averaged

monomeric free energy gained from various interactions and conforma-

tional constraints (e.g., hydrogen bonds, hydrophobic interactions, and

side-chain packing), kB is the Boltzmann constant, and T denotes the

temperature. The last term in Eq. 3 comes from the mixing entropy

of different filament species as in the Flory-Huggins theory (29). Note

the resemblance between the Boltzmann factors exp½�εn=ðkBTÞ�,exp½�εe=ðkBTÞ� and the two parameters s(for helix nucleation) and s (for

helix propagation) in the helix-coil transition theory (30), except that εnand εe are for b-structures (31). Furthermore, according to the well-known

relation between the Gibbs free-energy change for a reaction and the equili-

brium constant (32), we can correlate the energetic parameters in Eq. 3 and

the reaction rate constants introduced in Eq. 1 as εn ¼ �kBT lnðkþn =k�n Þ andεe ¼ �kBT lnðkþe =k�e Þ.

It is easily seen that the free energy thus defined is convex with respect to

the distribution ð½A1�; ½Anc �; ½Ancþ1�;.Þ. Therefore, we consider the minimi-

zation of the free energy under proper constraints:

min Fð½Ai�Þ (4)

XN XN XN

s:t:

i¼ nc

½Ai� ¼ P;i¼ nc

i$½Ai� ¼ M; ½A1� þi¼ nc

i$½Ai� ¼ mtot:

(5)

The first two constraints in Eq. 5 are based on the targeted macroquantities

PðtÞ and MðtÞ, and the last one is from the conservation law of mass.Because the free energy is convex, the above constrained optimization

problem can be solved simply by taking the variation

d

d½Ai�

"Fð½Ai�ÞðkBTÞ � l1

XNi¼ nc

½Ai� � P

!� l2

XNi¼ nc

i$½Ai� �M

!

� l3

½A1� þ

XNi¼ nc

i$½Ai� � mtot

!#¼ 0;

(6)

with Lagrangian multipliers l1; l2; l3 corresponding to three constraints.

From this, it follows directly that8><>:

½A1� ¼ expðl3Þ

½Ai� ¼ kþnk�n

�kþek�e

�i�nc

expðl1 þ il2 þ il3Þ: ðiRncÞ(7)

Moreover, it is not difficult to express the multipliers l1; l2; l3 in terms of

PðtÞ;MðtÞ;mtot:

l1 ¼ ln

�k�nkþn

P2

M � ðnc � 1ÞP� nc$ln

�k�ekþe

M � ncP

M � ðnc� 1ÞP;

l2 ¼ ln

�k�ekþe

M � ncP

M � ðnc � 1ÞP� ln½mtot �M�;

l3 ¼ ln½mtot �M�:

8>>>>>>><>>>>>>>:

(8)

Note that generally there is some freedom in the choice of free-energy func-

tion (Eq. 3) for the moment-closure method. Different forms of free energy

intrinsically correspond to different assumptions regarding the system (i.e.,

which kinds of interactions and motions are considered) and will lead to

different approximate fiber length distributions (Eqs. 7 and 8). In this

work, we choose the free energy according to Lee (33) and Schmit et al.

(34), which leads to a single exponential distribution of form ½Ai�fe�gi(g

is a constant). Fortunately, this choice accounts perfectly for the assumption

of fast equilibrium in the fiber elongation process. (There is a small differ-

ence between our free energy (Eq. 3) and Hill’s free energy (reformulated

with our notation):

F

ðkBTÞ ¼ � ln

kþfk�f

!XNi¼ nc

½Ai� � ln

�kþek�e

�XNi¼ nc

i$½Ai�

þ ð½A1� ln½A1� � ½A1�Þ þXNi¼ nc

ð½Ai� ln½Ai� � ½Ai�Þ

� nXNi¼ 1

ln i$½Ai�:

If we set kþn =k�n ðkþe =k�e Þ�nc ¼ kþf =k

�f , then the only difference from our

expression (Eq. 3) is the last term, nPN ln i$½Ai�. This term accounts for

i¼1

the translational and rotational freedoms of a fiber in solution. The latter

will lead to a distribution function of fiber length in the form of

½Ai�fine�gi corresponding to the partial equilibrium approximation on fiber

fragmentation process. Considering that elongation is usually much faster

than fragmentation in realistic amyloid fiber systems, our current free

energy is expected to give better results than Hill’s.)

Moment-closure equations

For the length-dependent reaction rate constants for filament fragmentation

and association,we refer toHill’s theoretical calculations (36) and take themas8>>>><>>>>:

kþf ði; jÞ ¼ kþf ðijÞn�1ði ln j þ j ln iÞðiþ jÞnþ1

;

k�f ði; jÞ ¼ k�f ði ln j þ j ln iÞ½ijðiþ jÞ� ;

(9)

where n represents the degrees of freedom of a filament in the solution. For

several typical amyloid fiber systems (see Fig. 3 D), we find n ¼ 1 � 3,

which means they are in a partially mobile state. (According to Hill’s argu-

ment (36), the value of index n accounts for the translational and rotational

degrees of freedom for a filament moving in solution. In principle, the

contribution from translation is 3/2, the contribution from two-dimensional

rotation of a rigid rod is 3, and the contribution from rotation about the axis

of a rigid rod is 1/2. Thus nz4 � 6 when a filament has completely free

motion in solution. On the contrary, nz0 for the immobile case. For several

typical amyloid fiber systems shown in Fig. 3 D, we found nz1 � 3. We

expect this is caused by some degrees of freedom for the filament motion

becoming frozen due to filaments cross-linking, bundling, and so on.

Thus, the filaments in these systems will lie in a state between freely mobile

and immobile, which we describe as partially mobile.)

Then, using the distribution function in Eq. 7, we derive our simple

model (or moment-closure equations) from Eq. 2 as

dP

dt¼ kþn ðmtot �MÞnc�k�n ð1� qÞP

þ kþf ð1� qÞPqncXn�1 � k�f ð1� qÞ2P2L1;

dM

dt¼ nck

þn ðmtot �MÞncþ2kþe ðmtot �MÞP� 2k�e P

� �nck�n � 2k�e�ð1� qÞP;

8>>>>>>>>>><>>>>>>>>>>:

(10)

Biophysical Journal 104(3) 533–540

536 Hong and Yong

where qhðM � ncPÞ=½M � ðnc � 1ÞP�˛½0; 1Þ. In Eq. 10, the functions

Xn�1 ¼ nnc2n

"ln nc

n2cð � ln qÞ2 þðn� 2Þ ln nc þ 1

n3cð � ln qÞ3

þ ð3n� 5Þðn� 4Þ ln nc þ 6ðn� 3Þ4n4cð � ln qÞ4 þ.

#

and

L1 ¼ ln nc

n2cð � ln qÞ2

are derived from the infinite summations in Eq. 2 through integration by

parts. In practice, to account for all positive contributions in the summa-

tions, only one term is needed for filament association (L1), and n� 1

terms (Xn�1) are required for filament fragmentation. Note that Xn�1 is

an asymptotic series and may diverge as q/1 (though ð1� qÞXn�1 still

converges).

To our knowledge, this is the first work to obtain closed-form equations

for the moments PðtÞ and MðtÞ when general length-dependent fiber frag-

mentation processes are considered. For models without fragmentation

(kþf ði; jÞ ¼ 0), exact solutions have been obtained by Oosawa and Asakura

(7), Lomakin et al. (38), and Cohen et al. (39). For models with length-inde-

pendent fragmentation (kþf ði; jÞ ¼ kþf ), closed-form equations for PðtÞ andMðtÞ can be derived by neglecting unimportant boundary terms. Further-

more, approximate solutions with high accuracy have been obtained

through fixed-point analysis (8,40,41). However, for models with length-

dependent fragmentation, self-closure of moment equations is usually an

illusion. In these cases, our moment-closure method provides a general

solution framework regardless of what kinds of real forms for fiber frag-

mentation are taken.

Relation with partial equilibrium approximation

The moment-closure method used here has a strong physical basis: the

minimization of the system free energy under given constraints (PðtÞ,MðtÞ, and mtot for the present case). Furthermore, we can show that this

moment-closure method is equivalent to the partial equilibrium approxima-

tion on fiber elongation in mathematics, and thus clarify the applicable

range of our method.

A notable aspect of the original mass-action equations is that the fiber

elongation processes alone satisfy the principle of detailed balance when-

ever they are reversible. In fact, for any given positive numbers ½A1�� and

½Anc ��, we can easily find positive numbers ½Ai�� ¼ ½Anc ��ðkþe ½A1��=k�e Þi�nc

(i>nc), such that

kþe ½A1��½Ai�� ¼ k�e ½Aiþ1��: (11)

Coincidently, with the constraintsPN

i¼nc½Ai�� ¼ P and

PNi¼nc

i$½Ai�� ¼ M,

the filament length distribution ½A �� ¼ qi�nc ð1� qÞP (iRn ) can be derived

i c

from Eq. 11. It is exactly the same as that obtained by the moment-closure

method in Eqs. 7 and 8. This connection suggests that in the current case,

the moment-closure method is equivalent to the partial equilibrium approx-

imation on fiber elongation.

To further discuss the mathematical foundation for the partial equilib-

rium approximation, we set XðtÞhð½A1�; ½Anc �; ½Ancþ1�;.ÞT . Then the

mass-action equations Eq. 2 can be rewritten into a vector form:

d

dtX ¼ 1

tnRnðXÞ þ 1

teReðXÞ þ 1

tfRf ðXÞ; (12)

where RnðXÞ, ReðXÞ, and Rf ðXÞ represent the terms from primary nucle-

ation, elongation, and fragmentation, respectively, and tn, te, and tf char-

Biophysical Journal 104(3) 533–540

acterize the timescales of the corresponding processes. In addition, one can

directly verify that ð0; 1; 1;.ÞReðXÞh0;ð1; nc; nc þ 1;.ÞReðXÞh0;

(13)

which means that PðtÞ and mtot are two conservative quantities for fiber

elongation.

Assume that the elongation processes are much faster than others, i.e., tnand tf are moderate and te ¼ t is small. Under this assumption, 1=te ReðXÞcan be regarded as a stiff term. Because elongation processes obey the

principle of detailed balance, we know from the singular perturbation

theory (42,43) that the solution Xt of Eq. 12 possesses the following

property: as t tends to zero, PtðtÞ ¼ ð0; 1; 1;.ÞXtðtÞ and mttotðtÞ ¼

ð1; nc; nc þ 1;.ÞXtðtÞ converge to the solutions of the following equationsuniformly for t in any given bounded time interval:8>>><>>>:

d

dtP ¼ ð0; 1; 1;.Þ

�1

tnRn

�X0�þ 1

tfRf

�X0�;

d

dtmtot ¼ ð1; nc; ncþ 1;.Þ

�1

tnRn

�X0�þ 1

tfRf

�X0�;

(14)

Here X0 is the unique solution to8<:

ReðXÞ ¼ 0;ð0; 1; 1;.ÞX ¼ P;ð1; nc; nc þ 1;.ÞX ¼ mtot:

(15)

To some extent, this argument provides a mathematical basis for the partial

equilibrium approximation, as well as the moment-closure method pro-

posed here. However, for general applications, the underlying connections

between the moment-closure method and partial equilibrium approximation

still need further clarification.

RESULTS

Numerical comparison and data fitting

According to our above mathematical argument, as long asthe reaction rate for fiber elongation is much larger than thatfor other processes (e.g., primary nucleation, fragmentation,and corresponding backward reactions), the moment-closure equations will provide a reasonable approximationin the calculation of PðtÞ and MðtÞ compared with the orig-inal mass-action equations. This is further confirmed by thenumerical calculations shown in Fig. 2, A and B. In contrast,large deviations in the filament length distribution can beobserved due to the presence of the fragmentation process(see Fig. 2 C). It is fair to say that Eq. 7 captures an averagetendency of the realistic filament length distribution,whereas the moving peaks in filament length distributionproduced by fragmentation are totally neglected. In prin-ciple, if more macroquantities (such as high-order momentsof filament length) are considered as constraints in theminimization problem, one can expect better results forthe filament length distribution. However, one must awarethat the corresponding computational complexity will bedramatically increased.

A B

C D

FIGURE 2 (A and B) Comparisons of mass-

action Eq. 2 (red circles), elongation-only model

(green dashed lines), and moment-closure

Eq. 10 (blue solid lines) in the calculation of PðtÞand MðtÞ with mtot ¼ 50mM, kþn ¼ 10�4M�1s�1,

kþe ¼ 5�103M�1s�1, kþf ¼ 5�10�7s�1, k�f ¼102M�1s�1, k�n ¼ k�e ¼ 0, nc ¼ 2, n ¼ 3. Three

major kinetic quantities are represented by black

dotted lines. (To highlight the significant roles of

fragmentation, we plot the results of the elonga-

tion-only model (38,39) (by simply neglecting the

fragmentation process kþf ði; jÞ ¼ 0) for compar-

ison. It can be clearly seen that the elongation-

only model is only applicable to the very initial

stage, in correspondence with the fact that elonga-

tion is much faster than fragmentation. Actually,

when the long-time behaviors are concerned, the

slow processes (fragmentation in this case) play

a major role. This is expected mathematically.

From a biological point of view, fragmentation

can provide more fiber seeds (even their reaction

rate constants look very small compared with other

processes), which will greatly affect the formation

of amyloid fiber.) (C) Comparison of exact fiber length distribution (calculated from Eq. 2 and shown by dots) and approximate fiber length distribution

(obtained in Eq. 7 and shown by solid lines) at different time. (D) Experimental data fitting for polymerization of the WW domain measured by Ferguson

et al. (44). Red circles indicate experimental data under different initial protein concentrations mtot ¼ 500; 200; 100; 50mM, respectively; blue solid lines

indicate numerical solutions of moment-closure equations (Eq. 11) with kþn ¼ 8� 10�9M�1s�1, kþe ¼ 1� 105M�1s�1, kþf ¼ 1:7� 10�13s�1,

k�f ¼ 1� 103M�1s�1, k�n ¼ k�e ¼ 0, nc ¼ 2, n ¼ 3. (Knowles et al. (8) performed a similar fitting for models with length-independent fragmentation.

However, their reaction rate constant for fragmentation appears to be much larger than ours due to their oversimplified assumption of length dependence.)

Moment-Closure Model for Amyloid Fiber 537

In contrast to the extremely long time required fora consistent calculation of the original mass-action equa-tions, our moment-closure equations offer great simplicityfor model analysis, numerical calculations, and experi-mental data fitting. In general, the performance is improvedby at least 10,000-fold (from days to seconds). In Fig. 2 D,we apply our simple model to study the polymerization dataof the WW domain measured by Ferguson et al. (44). Bychoosing appropriate reaction rate constants (the extractionof model parameters from the experimental data follows theprocedure described in Hong et al. (20)), we can fit all fourkinetic curves under different protein concentrations simul-taneously. This is a critical test of the moment-closure equa-tions (Eq. 10).

Basic model properties

We further explore the basic kinetic and equilibrium proper-ties of the moment-closure equations. To quantitativelycharacterize the kinetic curves for MðtÞ, two often-studiedquantities are the apparent fiber growth ratek1=2h _Mðt1=2Þ=mtot (defined as the normalized rate formass concentration changes at the half-time of fibrillationMðt1=2Þ ¼ mtot=2) and the lag time tlaght1=2 � 1=ð2k1=2Þ.Through mathematical analysis (at the half-time of fiberformation, we usually have M[P, thus, qz1 and�ln qz1� q; putting these relations into the first formulaof Eq. 2 and keeping the leading term of Xn�1, we getdP=dt � kþf M

n�1=Pn�2; combining the sigmoidal curve ofM with the definition of apparent fiber growth rate k1=2,

we have P � ðkþf =k1=2Þ1=ðn�1Þmtot; then, using the formuladM=dt � kþe mtotP, we get k1=2 � kþe ðkþf =k1=2Þ1=ðn�1Þmtot,which predicts the relation k1=2f½ðkþe mtotÞn�1kþf �1=n) andnumerical experiments, we find the following elegant rela-tions (see Fig. 3, A–C):

k1=2fh�kþe mtot

�n�1kþfi1=n

; (16)

h�þ�2=n � �i

tlagfln kf = kþn m

nc�1toth�

kþe mtot

�n�1kþfi1=n ; (17)

provided that the fragmentation is more efficient in produc-ing new seeds than primary nucleation at the half-timeof fiber formation ½ðkþe mtotÞn�1kþf �1=n[ðkþn kþe mnc

totÞ1=2,which can be easily verified. (This condition can beverified for all data sets used in Fig. 3 as follows: for

n ¼ 2, ½ðkþe mtotÞ1kþf �1=2¼10�6 �10�3=2[ðkþn kþe m2totÞ1=2¼

10�15 �10�9; for n ¼ 3, ½ðkþe mtotÞ2kþf �1=3 ¼ 10�7�10�2[ ðkþn kþe m2

totÞ1=2 ¼ 10�15 � 10�9; and for n ¼ 4,

½ ðkþe mtotÞ3 kþf �1=4 ¼ 10�27=4 � 10�3=2[ðkþn kþe m2totÞ1=2 ¼

10�15� 10�9). The above relations confirm that the modelwith constant filament fragmentation rates (8,20) can be re-garded as a special case of the length-dependent one with

n ¼ 2 (when n ¼ 2, kþf ði; jÞ=kþf � OðlnðiÞ þ lnðjÞÞ, thus it

can be roughly regarded as length independent).

Biophysical Journal 104(3) 533–540

A

C D

B FIGURE 3 (A–C) Scaling relationships among

apparent fiber growth rate, lag time, and model

parameters. For parameter n ¼ 2; 3; 4, 10,000

data points are generated separately with reaction

rate constants randomly chosen in mtot ¼0:1� 100mM, kþn ¼ 10�9 � 10�6M�1s�1, kþe ¼103 � 106M�1s�1, kþf ¼ 10�8 � 10�5s�1 for

n ¼ 2, kþf ¼ 10�13 � 10�10s�1 for n ¼ 3, kþf ¼10�15 �10�12s�1 for n ¼ 4, k�f ¼ k�n ¼ k�e ¼ 0,

nc ¼ 2. (D) Scaling relationship between lag

time and protein concentration (tlagfmatot). Data

are shown for Sup35 NW region (dark green

squares) (14), Ure2p (brown circles) (48),

CsgBtrunc (blue downward triangles) (49), Stefin

B (purple upward triangles) (50), b2-microglo-

bulin (pink dots) (51), WW domain (light green

crosses) (44), and insulin (red stars) (52). Black

dashed lines denote the best fitting curves for

each data set, with the slope numbered beside

the line. For scaling exponents a ¼ �0:3;

�0:4;�0:6, the corresponding model parameters

are n ¼ 1:4; 1:7; 2:5, respectively.

538 Hong and Yong

Furthermore, we see that the universal inverse relationship

k1=2ft�1lag between the apparent fiber growth rate and lag

time mentioned in the literature (47) holds only approxi-mately (see Fig. 3 C).

Note that the ðn� 1Þ=n-law (k1=2fmðn�1Þ=ntot ;

tlagfm�ðn�1Þ=ntot ) obtained above is different from the

ðnc þ 1Þ=2-law (7) for the classical nonbreakable filamentmodel (primary nucleation) and the ðn2 þ 1Þ=2-law (18)for the heterogeneous nucleation model (also nonbreak-able). Our simple model may provide an explanation forthe observed weak dependence (scaling exponent < 1)between the apparent fiber growth rate (or lag time) andinitial protein concentration for many amyloid proteins(8,20) (see also Fig. 3 D). More importantly, these differentscaling behaviors offer an effective way to extract the under-lying mechanisms from experimental data and choose thecorrect model for a given amyloid fiber system.

From Eqs. 16 and 17, we get a rough idea that the majorkinetic behaviors of MðtÞ are determined by three forwardreaction rate constants (primary nucleation, elongation,and fragmentation). However, when the equilibrium valuesof PðtÞ and MðtÞ are concerned, backward reaction rateconstants cannot be neglected. In fact, it is not difficult todeduce from our simple model (Eq. 10) that

mtot �MðNÞ � k�ekþe

; (18)

þ n�1!1=n

PðNÞ � kf mtot

k�f: (19)

Therefore, the nonzero value of k�e guarantees a measurableconcentration of monomers in the equilibrium (53), and the

Biophysical Journal 104(3) 533–540

presence of k�f is necessary for a reasonable equilibriumvalue of PðtÞ. If k�f ¼ 0, the predicted average length of fila-ments (� nc) will be much smaller than experimental values(from hundreds to tens of thousands) (20,42,43). This pointdid not receive enough attention in previous studies(8,54,55).

DISCUSSION

In addition to the two often-studied quantities, the zeroth-order moment PðtÞ and first-order moment MðtÞ, high-order moments have recently attracted much interest intheoretical analysis (54) and experimental measurements(e.g., the light intensity in static light scattering is propor-tional to the second-order moment of filament length(15)). High-order moments can be computed by using avail-able low-order moments; for example, in the current casethe qth-order moment can be computed as

PNi¼nc

iq$½Ai� ¼PNi¼nc

iq$qi�ncð1� qÞP. However, this method may not guar-antee good accuracy. Another approach is to introduce thetargeted high-order moments as new constraints in the mini-mization problem. Quantitative comparisons of twodifferent approaches are in progress.

An interesting question is, why does the moment-closuremethod work so well? Physically, the method is based on theprinciple of minimum free energy under certain constraints,so it seems reasonable. Mathematically, however, no theo-retical support is available in the literature. For the presentcase, we have pointed out the mathematical relationbetween the moment-closure method and the partial equilib-rium approximation, which to our best knowledge is the firstjustification for the mathematical correctness of thismethod. However, for general applications, it is still anopen problem.

Moment-Closure Model for Amyloid Fiber 539

Another interesting question is, how well does thisapproach compare with other methods? Wang et al. (56)compared several closure methods by means of numericalexperiments and claimed that the current method is thebest. However, for mass-action equations, quantitativeconclusions require further clarification.

This work was partially supported by the National Natural Science Founda-

tion of China (NSFC 10971113 and NSFC 11204150) and the Tsinghua

University Initiative Scientific Research Program (20121087902).

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